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YCTP-P7-99hep-th/9903416

Cosmological Neutrino Condensates

D.G. Caldi∗

Department of Physics, State University of New York at Buffalo,

Buffalo, NY 14260

Alan Chodos†

Department of Physics, Yale University, New Haven, CT 06520-8120,

USA

Abstract

We investigate the possibility that neutrinos form superfluid-type conden-

sates in background cosmological densities. Such condensates could give rise

to small neutrino masses and splittings, as well as an important contribution,

perhaps, to the cosmological constant. We discuss various channels in the

context of the standard model. Many of these do not support a condensate,

but some mixed-flavor channels do. We also suggest a new interaction, acting

only among neutrinos, that could induce a neutrino Majorana mass of order

1 eV.

PACS numbers:

∗Electronic address: caldi@caldi.physics.buffalo.edu

† Electronic address : chodos@hepvms.physics.yale.edu

For various reasons, but especially the recent evidence for neutrino oscillations and hence

neutrino mass [1], considerable effort has been devoted of late to neutrinos and their prop-

erties. Furthermore, another more speculative consideration has piqued interest in small

neutrino masses: a number of cosmological observations has led to renewed interest in the

long-discarded notion of a cosmological constant [2]. The preferred value of this constant

leads to a mass scale that is quite similar to the mass for neutrinos inferred from the at-

mospheric and solar neutrino observations, in the range of tenths of an eV to a few eV [1].

It seems to us that a natural framework in which to relate these seemingly disparate phe-

nomena would be the formation of some type of neutrino condensate, the energy of which

might provide the cosmological constant, while the neutrino masses could emerge from an

expansion about the symmetry-breaking vacuum.

Appropriate tools to investigate the possibility of neutrino pair condensation of the su-

perconductor type, have been recently employed to explore the possibility of qq condensates

in hadronic media at sufficiently high densities. This has been discussed in the context

of a four-fermi approximation to QCD, whether induced by instanton effects [3]or by one-

gluon exchange [4]. Similar effects have also been studied in the large-N approximation in

a 2-dimensional model related to the Gross-Neveu model [5].

For our purposes, we look for pairing phenomena in the electroweak theory. Specifically,

we suggest that condensates involving neutrinos may occur in cosmological situations where

the relevant chemical potentials are non-zero. Of course, the chemical potentials of the

various neutrinos in the universe are presumably quite tiny [6], reflecting as they do the

difference in density between neutrinos and anti-neutrinos of a given species. We shall

return to the estimation of the magnitude of the effects we discuss below; first we offer

a purely theoretical discussion of the following problem: to identify, if any, the attractive

channels in the electroweak theory that might permit neutrinos to condense. It should be

noted that the earliest speculation on this subject of which we are aware [7], took place

before the emergence of the standard model and the discovery of neutral currents, and the

authors simply assumed the existence of an attractive channel.

The hallmark of this type of pairing is that, once a Fermi surface forms, any attractive

channel, no matter how weak, can produce a condensate; this is in contrast to the more

familiar situation in hadronic physics of a chiral condensate, which typically requires the

coupling to exceed a certain threshold value.

The reason for this behavior of the pairing condensate can be understood in a variety

of ways. If one looks at the gap equation in a mean-field approximation one sees that, as

the gap tends to zero, a singularity develops at the Fermi surface; hence a non-zero gap is

1

necessary to prevent the formation of this singularity.

Alternatively, in a renormalization-group approach [8] one sees that the renormalized

coupling tends to infinity as one integrates out all modes above and below the Fermi surface;

thus even an apparently weak coupling grows without bound and allows the condensate to

form.

Our investigation will be limited to the leptonic sector of the electroweak theory. We

shall use the effective four-Fermi low-energy description of that theory and we shall perform

all our computations within the one loop (mean-field) approximation. Corrections to this

picture from the inclusion of effects not captured by the low-energy effective theory, or

from fluctuations about the mean-field approximation, must await further investigations.

We shall discuss possible consequences of extensions of the dynamics beyond the standard

model below.

To set the stage, let us consider an action with a generic four Fermi coupling:

S =1

2[ψ†

αAαβψβ − ψαATαβψ

†β ] + Mαβγδψ

†αψβψ

†γψδ .

Here the index α summarizes all the attributes of ψ: space-time dependence, gauge trans-

formation properties, flavor, spin and whatever else.

Because we are interested in pairings of the form 〈ψψ〉 and 〈ψ†ψ†〉, we assume that Mhas a Fierz-Bogoliubov (FB) decomposition:

Mαβγδ =∑

λ

ηλQ(λ)αγQ

∗(λ)βδ .

Here ηλ = ±1, and Q(λ)αγ = −Q(λ)

γα because of the Fermi statistics of ψ. Then we define

auxiliary fields B(λ), and add to L the term

−∑

λ

ηλ(B(λ)† −Q(λ)

αγψ†αψ

†γ)(B

(λ) +Q∗(λ)βδ ψβψδ) .

The original Lagrangian is recovered upon path-integrating on B(λ) and B(λ)†. With the

addition of this extra piece, the terms in L that are quartic in ψ and ψ† cancel, leaving us

with

2

S =1

2[ψ†Aψ − ψATψ†] −

∑

λ

ηλB†(λ)B(λ)

+ ψ†Bψ† + ψB†ψ ,

Bαγ =∑

λ

ηλB(λ)Q(λ)

αγ ,

B†βδ = −

∑

λ

ηλB†(λ)Q

∗(λ)βδ .

We obtain an effective action Γeff [B(λ), B(λ)†] by integrating over ψ and ψ†. The result is:

Γeff = −∑

λ

ηλB(λ)†B(λ) − i

2Trlog[1 + 4A−1B(AT )−1B†]

where we have dropped some terms independent of B and B†. The mean field approximation

that we shall adopt consists in demanding that Γ be stationary with respect to variations

of B and B†:

δΓ

δB(λ)=

δΓ

δB(λ)†= 0 .

These are the gap equations whose non-trivial solutions, if any, determine whether a

condensate with the quantum numbers of B(λ) and B(λ)† can form. Because the effective

theory we are dealing with is non-renormalizable, it is necessary to regularize these gap

equations, and the size of the gap depends, apparently, on the value of the cutoff. Thus

the gap equations in the form we derive them are not a good guide to the size of the gap.

However, they allow us to determine which channels are attractive and hence in which

channels one might expect a condensate to form. We shall see explicitly how this works in

particular cases below.

The simplest case is only one flavor of neutrino interacting with itself via the neutral

current. Since vector exchange produces repulsion among like particles, we intuitively expect

the ν−ν channel to be repulsive in this case. This is borne out by our explicit computation,

which we now describe.

The starting point is the four-Fermi interaction

Lint = −G2ψγµψψγµψ

where γ5ψ = −ψ. The sign is dictated by the standard model. If we use 2-component

fermions, and make use of the identity∑

a σaαβσ

aγδ = 2[ǫαγǫδβ + 1

2δαβδγδ] we see that

3

Lint = 2G2(ψ†αǫαγψ

†γ)(ψβǫβδψδ) . (1)

Making contact with our previous notation, we see that there is only one term in the

FB decomposition with Qαγ =√

2Gǫαγ , and η = −1. We remark that the fact that η = −1

already tells us that the interaction is repulsive at the tree level; it is then to be expected

that the gap equation will not have a non-trivial solution.

For the kinetic part of the action, again using 2-component notation, we have A =

i ∂∂t− i~σ · ~▽− µ, where µ is the chemical potential. Hence

A−1 =∫

d4p

(2π)4[

(p0 − µ) − ~p · ~σ(p0 − µ+ iǫsgnp0)2 − ~p2

]e−ip·(x−y)

and

(A−1)T = −∫

d4p

(2π)4

[(p0 + µ) − ~p · ~σT ]

(p0 + µ+ iǫsgnp0)2 − ~p2e−ip·(x−y) .

Note that the iǫ prescription has been introduced in the appropriate manner to take

account of the role of µ as the chemical potential. One then proceeds to construct

X ≡ 4A−1B(AT )−1B† which, under the assumption (appropriate for a vacuum solution)

that B and B† are constants, can be written

X = −∫

d4p

(2π)4F(p)e−ip·(x−y)

where

F(p) =+8G2B†B

p20 − (~p · ~σ − µ)2 + iǫ

.

The gap equation will involve Tr(1 + X)−1X. Doing the p0 integral in this expression

provides a factor of i that cancels the explicit i appearing in Γeff . The remaining integral

over ~p is ultraviolet divergent. We regularize this by imposing a cutoff Λ on the magnitude

of ~p. This leads to the unrenormalized gap equation, written in terms of M = B†B

1 =−G2

π2

∫ Λ

−Λdpp2 1

√

(p− µ)2 + 8MG2.

Clearly there is no solution (whereas there would have been a solution if we had had η = 1;

this would have changed the sign of the l-h s). Note that for M = 0, there is a logarithmic

divergence coming from p = µ. This is the singularity at the Fermi surface mentioned above.

4

Including an arbitrary number N of flavors turns out to be fairly straightforward assum-

ing no flavor mixing. One begins with the analog of eqn. (1):

Lint = 2G2(ψ†(i)α ǫαγψ

†(j)γ )(ψ

(i)β ǫβδψ

(j)δ ) (2)

where i, j are flavor indices summed from 1 toN , and performs a separate FB transformation

on the flavor indices using the identity

2δikδjl =2

Nδijδkl +

N2−1∑

a=1

(λa)ij(λa)∗kl

where the λ’s are the generators of the fundamental representation of SU(N), normalized

such that Trλaλb = 2δab.

Following an analysis paralleling the one-flavor case,‡ one finds

X ijαβ(x− y) = −Mij

∫

d4p

(2π)4Fαβ(p)e−ip·(x−y)

where

F =4G2

p20 − (~p · ~σ − µ)2 + iǫ

and

Mij =∑

A,B

B(A) B(B)† λ(A)ik λ

(B)kj .

Here the summation runs over the symmetric λ’s, including the unit matrix suitably nor-

malized. It is important to note that M is a positive matrix: M = KK†, K =∑

AB(A)λ(A).

This expression for X leads to the following set of gap equations:

B(A) =−G2

2π2B(B)

∫ Λ

−Λdp p2tr[

1√

(p− µ)2 + 4G2Mλ(B)λ(A)] .

Multiply by B(A)† and sum on A. This produces

‡In this analysis, we assume a common chemical potential for all the flavors. What may happen

if this is not the case will be discussed briefly below.

5

∑

A

B(A)† B(A) =−G2

2π2

∫ Λ

−Λdp p2tr[

1√

(p− µ)2 + 4G2MM ]

whose only solution is B(A) = 0 for all A.

Thus, if we confine ourselves to neutrinos alone, and to the dynamics of the standard

model, we find no possibility of neutrino pairing. Among the ways to avoid this conclusion

are (a) extend the dynamics beyond the standard model (we shall discuss this possibility

in the conclusions); (b) enlarge the dynamics to include the charged leptons (and possibly

also the quarks). We have performed an analysis in which we consider not only neutrinos

themselves but also electrons circulating in the loop. This generates an additional term in

Γeff , and hence an additional contribution to the gap equation for the neutrino condensate.

It does not, however, alter the result that there is no solution to the gap equation; (c)

Finally, we can consider condensates that are composed not of neutrinos alone, but that pair

neutrinos with charged leptons. Of course, since these condensates would be charged, their

phenomenological consequences would be much more drastic than those of purely neutrino

condensates.

In any event, we begin with the Lagrange density L = L0 + Lint, where

L0 = e(i▽/−m)e+ νei▽/ νe + νµi▽/ νµ − µee†e

− µνν†eνe − µ′

νν†µνµ .

Lint =−g2

8m2W

[νeγµ(1 − γ5)eeγµ(1 − γ5)νe −

1

2νeγµ(1 − γ5)νeeγ

µ(1 − γ5)e

+ 2sin2θW eγµeνeγµ(1 − γ5)νe −

1

2νµγµ(1 − γ5)νµeγ

µ(1 − γ5)e

+ 2sin2θW eγµeνµγµ(1 − γ5)νµ] .

We have kept only the electron, as by far the lightest charged lepton. We have exhibited

its couplings to νe and νµ; there should also be ντ , which we ignore for simplicity. It enters

in an identical manner to νµ.

Now we must implement the FB transformation. It is convenient first to make a Fierz

transformation on the charged-current terms, to get them in a form resembling the neutral

current terms, and then to make a FB transformation on the result. We find

Lint =g2

8m2W

[4sin2θW (νeγµeνeγµe+ νµγµeνµγ

µe)

+ 4(1 + 2sin2θW )νeγ0γ2eνeγ

0γ2e

+ 4(−1 + 2sin2θW )νµγ0γ2eνµγ

0γ2e] .

6

Note that γ0γ2 is the charge conjugation matrix, and hence expressions of the form νγ0γ2e

are proper Lorentz scalars. If we assume that the condensate will be a Lorentz scalar (this is

only for convenience: there is no particular reason to expect that these condensates should

respect Lorentz symmetry) then we limit ourselves to the last 2 terms. Note that they are

of opposite sign. It turns out that the term involving νe is ”bad” (η = −1) whereas the

one involving νµ is ”good” (η = +1). Hence we concentrate only on the last term, and

introduce an auxiliary field B for it. We let κ2 = g2

2M2

W

(1 − 2sin2θW ), and for notational

convenience we introduce a doublet ψ = (e, νµ) (a peculiar object from the point of view

of the underlying standard model), so that, for example, mee = m2ψ(1 + τ3)ψ and µee

†e +

µ′νν

†µνµ = ψ†(µ1 +µ2τ3)ψ. In our earlier notation, we have A = γ0(i▽/− m

2(1+τ3))−µ1−µ2τ3

and B = −κ2

4B(1 − γ5)γ

2γ0τ1.

Following the same analytic path as before, we arrive at a somewhat more complicated

gap equation:

−κ2B†B = −iκ4B†B

π3

∫ ∞

−∞dp p2

∫ ∞

−∞dp0G(p0, p)

where

G(p0, p) =p0 − µ+ + p

[(p0 − µ+)2 − p2 −m2][p0 + µ− + p] − 4κ4B†B[p0 − µ+ + p].

Here µ± = µ1 ± µ2, and the integration over poles on the real p0 axis is by the prescription

p0 → p0 + iǫsgnp0.

It is not possible to make much analytic headway with the expression in the general case.

But if we set △2 = 4κ4B†B = 0, the expression becomes tractable, and we can then address

two questions: (i) does the integral have the correct sign to permit a solution of the gap

equation? and (ii) is there an infrared singularity, which would be evidence of an instability

that could be cured by setting △2 6= 0? Let ω =√p2 +m2 > 0. Then the p0 integral can

be done, yielding

1 =κ2

π2

∫ Λ

−Λdp p2{θ(ω

2 − µ2+)

ω[(ω − p)θ(−p− µ−)

ω − p− 2µ1

+(ω + p)θ(p + µ−)

ω + p+ 2µ1

]

+4µ1

(ω + p+ 2µ1)(ω − p− 2µ1)[θ(−µ+ − ω)θ(−p− µ−) − θ(µ+ + ω)θ(p+ µ−)]} .

Because of the θ-functions, each term in the integrand is non-negative, thereby answering

the first equation in the affirmative.

7

Further, one sees that singularities can occur only at the Fermi momentum, p2F = µ2

−, and

then only if the condition µ2+−m2 = µ2

− is met. Since the electron density is proportional to

[µ2+ −m2]3/2 while the neutrino density is proportional to µ3

− [9], this amounts to equating

the Fermi momenta of the members of the pair. Phenomenologically, therefore, it is unlikely

that this type of condensate could occur, except maybe in the early universe when larger

background densities of both electrons and neutrinos were present.

We turn now to a consideration of the size of the condensate. A simple BCS-like estimate

gives

∆ ∼ pFe− 1

p2

FG2

.

If the G2 in eq. (1) or (2) is of order GF , then, given the allowed range of pF , the

exponential suppression makes ∆ very small. One is led, therefore, to suggest the existence

of a new interaction, acting only on neutrinos, for which the effective G2 would have a scale

of approximately 1 eV instead of the 200 GeV characteristic of GF . An interaction of the

same form as eqs. (1) or (2) would serve nicely, provided only we change the sign, thereby

generating an attractive channel. The condensate would then produce Majorana neutrino

masses of the form mν ∼ ∆ = G2〈νν〉, and possibly a contribution to the cosmological

constant Λ ∼ G2 | 〈νν〉 |2. Since both G and the fermi momentum pF are of order 1 eV, one

obtains neutrino masses and a cosmological constant that are likewise of this order.

Furthermore, if we generalize our earlier analysis and allow the chemical potentials for

the different neutrino species to vary, the condensates could depend non-trivially on flavor,

perhaps leading to an interesting spectrum of neutrino masses and mixings.

Our conclusions can be enumerated as follows:

(i) There is no attractive channel in the purely neutrino sector of the standard model;

(ii) The addition of charged leptons leads to attraction in the flavor off-diagonal channels,

but a pairing instability occurs only if the Fermi momenta of the neutrino and the charged

leptons are equal;

(iii) In the case of neutrino-charged lepton pairing, there may also be the possibility of

condensation in a Lorentz non-invariant channel. We have not looked at this in detail;

(iv) If a new interaction exists among neutrinos with characteristic scale 1 eV, neutrino

condensates could form with the right size to generate an interesting spectrum of masses

and mixings, as well as an appropriate contribution to the cosmological constant. This

possibility is currently under active investigation.

8

Acknowledgements

We wish to thank Fred Cooper, Gregg Gallatin, James Hormuzdiar and Hisakazu Mi-

nakata for interesting discussions. The work of AC was supported in part by U.S. Depart-

ment of Energy Grant #DE-FG02-92ER-40704. The work of DGC was supported in part

by U.S. Department of Energy Grant #DE-FG02-92ER-40741.

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